The operator \(M_\phi \)  is a bounded linear operator on \(L^2 (X; \mathcal {H})\)  satisfying \(\Vert M_\phi \Vert =\Vert \phi \Vert _{L^\infty }\)  provided \(X\)  is a \(\sigma \) -finite measure space. If \(X\)  is not a \(\sigma \) -finite measure space, then \(\Vert M_\phi \Vert \)  need not be same as \(\Vert \phi \Vert _{L^\infty }\)  [1]}. We define Parseval determining set for \(L^1 (X)\) , introduced by Iversion [2]} which is a measure-theoretic abstraction of characters for a locally compact abelian group.
